# Construct Triangle by Angle, Median, and Circumradius

### What Might This Be About?

### Problem

Construct $\Delta ABC,$ given the median $m_b,$ the circumradius $R,$ and the angle $\alpha$ at $A.$

### Construction

**Note:** The construction, as presented, is only valid when $\alpha$ is acute.

**Step 1**: Construct circle $(O)$ of radius $R$ and a right triangle with hypotenuse on the diameter and one angle $\alpha.$ By the Law of Sines the side of the triangle opposite $\alpha$ will be equal to side $a$ of the sought triangle. We take the end points of that segment as $B$ and $C.$

**Step 2**: Draw circles $C(B,m_b),$ centered at $B$ with radius $m_b,$ and $C(C,R/2)$ centered at $C$ with radius $R/2.$ Their intersections serve as $M_b,$ the foot of the median from $B.$ There could be $0,$ $1,$ or $2$ intersections.

**Step 3**: Find $A$ at the intersection of $(O)$ and $CM_b.$

### Acknowledgment

The construction is due to Prof. Dr. René Sperb.

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